{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T11:41:54Z","timestamp":1780659714907,"version":"3.54.1"},"reference-count":34,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,6]],"date-time":"2022-09-06T00:00:00Z","timestamp":1662422400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The idea of symmetry is a built-in feature of the metric function. In this paper, we investigate the existence and uniqueness of a fixed point of certain contraction via orthogonal triangular \u03b1-orbital admissible mapping in the context of orthogonal complete Branciari metric spaces endowed with a transitive binary relation. Our results generalize and extend some pioneering results in the literature. Furthermore, the existence criteria of the solutions to fractional integro-differential equations are established to demonstrate the applicability of our results.<\/jats:p>","DOI":"10.3390\/sym14091859","type":"journal-article","created":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T09:51:09Z","timestamp":1662630669000},"page":"1859","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Common Fixed-Points Technique for the Existence of a Solution to Fractional Integro-Differential Equations via Orthogonal Branciari Metric Spaces"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6346-605X","authenticated-orcid":false,"given":"Arul Joseph","family":"Gnanaprakasam","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, SRM Nagar, Kattankulathur 603203, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gunasekaran","family":"Nallaselli","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, SRM Nagar, Kattankulathur 603203, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Absar Ul","family":"Haq","sequence":"additional","affiliation":[{"name":"Department of Natural Sciences and Humanities, University of Engineering and Technology (Narowal Campus), Lahore 54000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4191-3338","authenticated-orcid":false,"given":"Gunaseelan","family":"Mani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8941-2527","authenticated-orcid":false,"given":"Imran Abbas","family":"Baloch","sequence":"additional","affiliation":[{"name":"Higher Education Department, Government Graduate College for Boys Gulberg, Punjab 54660, Pakistan"},{"name":"Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sur les operations dans les ensembles abstraits et leur application aux equations integrales","volume":"3","author":"Banach","year":"1922","journal-title":"Fund. Math."},{"key":"ref_2","first-page":"31","article-title":"A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces","volume":"57","author":"Branciari","year":"2000","journal-title":"Publ. Math."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Turinici, M. (2012). Functional contractions in local Branciari metric spaces. arXiv.","DOI":"10.1007\/s11565-012-0164-6"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"339","DOI":"10.37193\/CJM.2015.03.10","article-title":"Some fixed points results on Branciari metric spaces via implicit functions","volume":"31","author":"Karapinar","year":"2015","journal-title":"Carpathian J. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2154","DOI":"10.1016\/j.na.2011.10.014","article-title":"Fixed point theorems for \u03b1-\u03c8-contractive type mappings","volume":"75","author":"Samet","year":"2012","journal-title":"Nonlinear Anal. Theor."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1186\/1687-1812-2014-190","article-title":"Some new fixed point theorems for \u03b1-Geraghty contraction type maps in metric spaces","volume":"2014","author":"Popescu","year":"2014","journal-title":"Fixed Point Theory Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1186\/1029-242X-2014-38","article-title":"A new generalization of the Banach contraction principle","volume":"2014","author":"Jleli","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1186\/1029-242X-2014-439","article-title":"Further generalizations of the Banach contraction principle","volume":"2014","author":"Jleli","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1186\/1687-1812-2013-94","article-title":"An (\u03b1-\u03c8)-Meir-Keeler contractive mappings","volume":"2013","author":"Karapinar","year":"2013","journal-title":"Fixed Point Theory Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1186\/1687-1812-2013-129","article-title":"Generalized metrics and Caristi\u2019s theorem","volume":"2013","author":"Kirk","year":"2013","journal-title":"Fixed Point Theory Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"141409","DOI":"10.1155\/2015\/141409","article-title":"Fixed points results for \u03b1-admissible mapping of integral type on generalized metric spaces","volume":"2015","author":"Karapinar","year":"2015","journal-title":"Abstr. Appl. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1186\/s13660-016-1010-7","article-title":"Generalized contractions with triangular \u03b1-orbital admissible mapping on Branciari metric spaces","volume":"2016","author":"Arshad","year":"2016","journal-title":"J. Inequal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"569","DOI":"10.24193\/fpt-ro.2017.2.45","article-title":"On orthogonal sets and Banach fixed point theorem","volume":"18","author":"Gordji","year":"2017","journal-title":"Fixed Point Theory"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1007\/s11784-016-0297-9","article-title":"Orthogonal sets: The axiom of choice and proof of a fixed point theorem","volume":"18","author":"Baghani","year":"2016","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_15","first-page":"251","article-title":"Fixed point theory in generalized orthogonal metric space","volume":"6","author":"Habibi","year":"2017","journal-title":"J. Linear Topol. Algeb."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"505","DOI":"10.5937\/KgJMath1804505S","article-title":"New fixed point results in orthogonal metric spaces with an application","volume":"42","author":"Senapatt","year":"2018","journal-title":"Kragujev. J. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1007\/s11784-019-0737-4","article-title":"Fixed point theorems for orthogonal F-contraction mappings on O-complete metric spaces","volume":"21","author":"Sawangsup","year":"2020","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_18","first-page":"6692112","article-title":"Fixed point of orthogonal F-Suzuki contraction mapping on O-complete b-metric spaces with applications","volume":"2021","author":"Beg","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_19","first-page":"1198","article-title":"Solving a nonlinear integral equation via orthogonal metric space","volume":"7","author":"Gunaseelan","year":"2022","journal-title":"AIMS Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"481","DOI":"10.31801\/cfsuasmas.970219","article-title":"Some fixed point theorems on orthogonal metric spaces via extensions of orthogonal contractions","volume":"71","author":"Gungor","year":"2022","journal-title":"Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat."},{"key":"ref_21","first-page":"7251823","article-title":"Solving an Integral Equation via Orthogonal Branciari Metric Spaces","volume":"2022","author":"Aiman","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2859","DOI":"10.1016\/j.na.2009.11.029","article-title":"On the concept of solution for fractional differential equations with uncertainty","volume":"72","author":"Agarwal","year":"2010","journal-title":"Nonlinear Anal. Thoer."},{"key":"ref_23","first-page":"561","article-title":"The application of meir-keeler condensing operators to a new class of fractional differential equations involving \u03c8-Caputo fractional derivative","volume":"5","author":"Baitiche","year":"2021","journal-title":"J. Nonlinear Var. Anal."},{"key":"ref_24","first-page":"20120144","article-title":"Some existence results on nonlinear fractional differential equations","volume":"371","author":"Baleanu","year":"2013","journal-title":"Philos. Trans. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"804","DOI":"10.1006\/jmaa.2000.7123","article-title":"The existence of a positive solution for nonlinear fractional differential equation","volume":"252","author":"Zhang","year":"2000","journal-title":"J. Math. Anal. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/S0022-247X(02)00583-8","article-title":"Existence of positive solutions for some class of nonlinear fractional equation","volume":"278","author":"Zhang","year":"2003","journal-title":"J. Math. Anal. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1140","DOI":"10.1016\/j.cnsns.2010.05.027","article-title":"Recent history of fractional calculus","volume":"16","author":"Machado","year":"2011","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_28","unstructured":"Lakshmikantham, V., and Rao, M.R.M. (1995). Theory of Integro-Differential Equations, CRC Press."},{"key":"ref_29","first-page":"107","article-title":"Some existence results for fractional integro-differential equations with nonlinear conditions","volume":"12","author":"Ahmad","year":"2008","journal-title":"Commun. Appl. Anal."},{"key":"ref_30","first-page":"23","article-title":"Stability analysis for a generalized proportional fractional Langevin equation with variable coefficient and mixed integro-differential boundary conditions","volume":"2020","author":"Sudsutad","year":"2020","journal-title":"J. Nonlinear Funct. Anal."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"6615478","DOI":"10.1155\/2020\/6615478","article-title":"Multivalued problems, orthogonal mappings, and fractional integro-differential equation","volume":"2020","author":"Sharma","year":"2020","journal-title":"J. Math."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Acar, \u00d6., and \u00d6zkapu, A.S. (2022). Multivalued rational type F-contraction on orthogonal metric space. Math. Found. Comput.","DOI":"10.15330\/cmp.14.2.485-492"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3125","DOI":"10.3934\/math.2020201","article-title":"Fixed point theorems in R-metric spaces with applications","volume":"5","author":"Khalehoghli","year":"2020","journal-title":"AIMS Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s40096-020-00338-5","article-title":"R-topological spaces and SR-topological spaces with their applications","volume":"14","author":"Khalehoghli","year":"2020","journal-title":"Math. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1859\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:24:21Z","timestamp":1760142261000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1859"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,6]]},"references-count":34,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["sym14091859"],"URL":"https:\/\/doi.org\/10.3390\/sym14091859","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,6]]}}}